S.Guilloteau (IRAM/ESO) ALMA European Project Scientist · S.Guilloteau (IRAM/ESO) ALMA European Project Scientist Bibliography: • “Synthesis Imaging”. ... Basic principle •

Introduction to millimeter Interferometry

S.Guilloteau (IRAM/ESO)

ALMA European Project Scientist

Bibliography:

• “Synthesis Imaging”. Proceedings of the lectures from NRAO summer school. EdsR.Perley, F.Schwab & A.Bridle.

• “ Proceedings of the IMISS2”. Ed. A.Dutrey

• “Interferometry and Synthesis in Radio Astronomy”. R.Thompson, J.Moran &G.W.Swenson, Jr.

• “Radio Interferometry”, S.Guilloteau, in “High Angular Resolution in Astrophysics”p154-163, A.M.Lagrange et al. (eds)

ESO - Chile / January 2002 Introduction to mm interferometry 1

Why Millimeter ?

• This is the next best atmospheric window longward of theIR cutoff.

•Most of the Universe is cold (today).

• Emission from a black body at 10 K peaks in the mmdomain.

• The most abundant molecules (e.g. CO, HCN) have theirfundamental rotation lines in the mm domain.(*)Exceptions: the lighter hydrides have their fundamental athigher frequencies, and more complex, heavier, moleculeshave their fundamental at longer wavelengths)

• The low temperature also implies the peak of the emissionfrom molecules (even heavy ones) is in the mm domain.


Why Interferometry ?

• Angular resolution goes (only) as λ/D

• To get 1′′ at 1 mm, we need D = 200 m

• As far as I know, it is somewhat difficult to build a fullysteerable football stadium

• Specially if it has to be (blindly) oriented within 0.5′′

precision• So we have to rely on the Young’s holes...


Basic principle

• The antenna produces an output Voltage proportional to the linear superposition ofthe incident electric field pattern. For a simple monochromatic case:

U(t) = Ecos(2πνt + φ) (1)

• In the receiver, a mixer superimposes the field generated by a local oscillatorto the antenna output.

ULO(t) = Qcos(2πνLOt + φLO) (2)

• The mixer is a non-linear element (diode) whose output is

I(t) = a0 + a1(U(t) + ULO(t)) + a2(U(t) + ULO(t))2 + a3(U(t) + ULO(t))3 + ... (3)

The second order (quadratic) term of Eq.3 can be expressed as

I(t) = a2E2 cos2(2πνt + φ) + 2a2EQ cos(2πνt + φ) cos(2πνLOt + φLO)

+ a2Q2cos2(2πνLOt + φLO) (4)


• Developing the product of the two cosine functions, we obtain

I(t) = ... + a2EQcos(2π(ν − νLO)t + φ− φLO) + ... (5)

There are obviously other terms in ν + νLO, 2νLO, 2ν, 3νLO ± ν, etc. . . in the aboveequation, as well as terms at very different frequencies like ν, 3ν, etc. . .

• By inserting a filter at the output of the mixer, we can select only the term such that

νIF −∆ν/2 ≤ |ν − νLO| ≤ νIF + ∆ν/2 (6)

where νIF , the so-called Intermediate Frequency, is a frequency which issignificantly different from the original signal frequency ν (which is often called theRadio Frequency νRF ).

• Hence, after mixing and filtering, the output of the receiver is

I(t) ∝ EQcos(±(2π(ν − νLO)t + φ− φLO)) (7)


• The receiver output is

I(t) ∝ EQcos(±(2π(ν − νLO)t + φ− φLO)) (8)

– changed in frequency: ν → ν − νLO or ν → νLO − ν

– proportional to the original electric field of the incident wave: ∝ E

– with a phase relation with this electric field:φ → φ− φLO or φ → φLO − φ

– proportional to the local oscillator voltage: ∝ Q

• The frequency change, usually towards a lower frequency, allows to select νIF suchthat amplifiers and transport elements are easily available for further processing.

•

Relation between the IF, RF and local oscillator frequencies in an heterodyne system


• The mixer described before accepts simultaneously frequencies which are

– higher than the local oscillator frequency: this is called Upper Side Band(USB) reception

– lower than the local oscillator frequency: this is called Lower Side Band(LSB) reception

and cannot a priori distinguish between them. This is called Double Side Band (DSB)reception.

• Some receivers are actually insensitive to one of the frequency range, either because afilter has been placed at the receiver input, or because their response is very stronglyfrequency dependent. Such receivers are called Single Side Band (SSB) receivers.

• An important property of the receiving system expressed by Eq.?? is that the sign ofthe phase is changed for LSB conversion.

• This property can be easily retrieved recognizing that the Frequency ν is the timederivative of the Phase φ.


The Heterodyne Interferometer

Schematic diagramof the two-antennaradio interferometer.τg = b.s/c is calledthe geometrical delay

Let us forget the frequency conversion for some time, i.e. assume νIF = νRF ...


• The input (amplified) signals from 2 elements of the interferometer are processed bya correlator, which is just a voltage multiplier followed by a time integrator.

• With one incident plane wave, the output r(t) is

r(t) =< v1 cos(2πν(t− τg(t)))v2 cos(2πνt) >= v1v2 cos(2πντg(t)) (9)

where τg is the geometrical delay τg(t) = (b.s)/c

• As τg varies slowly because of Earth rotation, r(t) oscillates as a cosine function, andis thus called the fringe pattern. As we had shown before that v1 and v2 wereproportional to the electric field of the wave, the correlator output (fringe pattern) isthus proportional to the power of the wave.

•

Position vectors used for the ex-pression of the interferometer re-sponse to an extended source,schematically represented by theiso-contours of the sky brightnessdistribution.


Source Size Effects• The signal power received from a sky area dΩ in direction s is (see Fig. for notations)

A(s)I(s)dΩdν

over bandwidth dν, where A(s) is the antenna power pattern (assumed identical forboth elements, more precisely A(s) = Ai(s)Aj(s) with Ai the voltage pattern ofantenna i), and I(s) is the sky brightness distribution

dr = A(s)I(s)dΩdν cos(2πντg) (10)

r = dν

∫Sky

A(s)I(s) cos(2πνb.s/c)dΩ (11)

• Two implicit assumptions have been made in deriving Eq.11.

– We assumed incident plane waves, which implies that the source must be in thefar field of the interferometer.

– We used a linear superposition of the fringes from the incident waves, whichimplies that the source must be spatially incoherent.

• These assumptions are quite valid for most astronomical sources, but may be violatedunder special circumstances (e.g. VLBI observations of solar system objects wouldviolate the first assumption, or masers for the second one.)


• When the interferometer is tracking a source in direction so, with s = so + σ

r = dν cos(2πνb.so

c)

∫Sky

A(σ)I(σ) cos(2πνb.σ/c)dΩ

− dνsin(2πνb.so

c)

∫Sky

A(σ)I(σ) sin(2πνb.σ/c)dΩ (12)

• We define the Complex Visibility

V = |V |eiφV =

∫Sky

A(σ)I(σ)e(−2iπνb.σ/c)dΩ (13)

which resembles a Fourier Transform...

• We thus have

r = dν cos(2πνb.so

c)|V | cos(φV )− dν sin(2πν

b.so

c)|V | sin(φV )

= dν |V | cos(2πντg − φV ) (14)

i.e. the correlator output is proportional to the amplitude of the visibility, andcontains a phase relation with the visibility.


Finite Bandwidth

• Integrating over dν,

R =1

∆ν

∫ ν0+∆ν/2

ν0−∆ν/2|V | cos(2πντg − φV )dν (15)

• Using ν = ν0 + n

R =1

∆ν

∫ ∆ν/2

−∆ν/2|V | cos(2πν0τg − φV + 2πnτg)dn (16)

= |V | cos(2πν0τg − φV )sin(π∆ντg)

π∆ντg

(17)

• The fringe visibility is attenuated by a sin(x)/x envelope, called the bandwidthpattern, which falls off rapidly. 1% loss in visibility <==> |∆ντg| ' 0.078, or with∆ν = 500MHz and a baseline length b = 100m, when the zenith angle is 2 arcminonly.

• The ability to track a source for a significant hour angle coverage requires propercompensation of the geometrical delay when a finite bandwidth is desired.


Delay Tracking and Frequency Conversion• Delay lines with mirrors (as in optics...) are impractical: avoiding diffraction losses

requires D2m Bλ, i.e. Dm 9 m for ALMA (λ = 7 mm, B = 12 km).

• The compensating delay must be introduced after one (or several) frequencyconversion(s)

•2-element heterodyne in-terferometer with delaytracking after frequencyconversion


• For USB conversion, the phase changes of the input signals before reaching thecorrelator are

φ1 = 2πντg = 2π(νLO + νIF )τg (18)

φ2 = 2πνIFτi + φLO (19)

since the delay is inserted at frequency νIF

• ∆τ = τg − τi being the delay tracking error, the correlator output is

r = |V | cos(φ1 − φ2 − φV )

rUSB = |V | cos(2π(νLOτg + νIF∆τ )− φV − φLO)

rLSB = |V | cos(2π(νLOτg − νIF∆τ )− φV − φLO)

(20)

• When the two sidebands are superposed,

rDSB = 2|V | cos(2π(νLOτg − φV − φLO)) cos(2πνIF∆τ ) (21)

i.e. the amplitude is modulated by the delay tracking error.

• We use sideband separation to avoid this problem. The delay tracking error shouldthen be kept small compared to the bandwidth ∆τ 1/∆ν.


Fringe Stopping

• With the Earth rotation, the cosine term of Eq.17 modulates the correlator outputwith a natural fringe rate of

νLO

dτg

dt' Ωearth

bνLO

c(22)

which is of order of 10 Hz for b= 300 m baselines and νLO = 100 GHz, or 2′′ angularresolution (since the fringe rate only depends on the effective angular resolution).

• Fringe rate too high for digital sampling of the visibility. Exception: VLBI, althoughresolutions are < 1 mas.

• Usual technique: modulate the phase of the local oscillator φLO such that

φLO(t) = 2πνLOτg(t)

at any given time. Then is a slowly varying output, which would be constant for apoint source at the reference position (also called the delay tracking center). Thisprocess is called Fringe Stopping.


Complex Correlator

• After fringe stoppingrr = Ao|V |cos(±2πνIF∆τ − φV ) (23)

we can no longer measure the amplitude |V | and the phase φV separately.

• A second correlator, with one signal phase shifted by π/2 becomes necessary. Itsoutput is

ri = Ao|V |sin(±2πνIF∆τ − φV ) (24)

• With both correlators, we measure directly the real rr and imaginary ri parts of thecomplex visibility r. The device is thus called a complex correlator.

• Note: A delay tracking error ∆τ appears as a phase slope as a function offrequency, with

φ(νIF ) = ±2πνIF∆τ (25)

with the + sign for USB conversion, and the – sign for LSB conversion.


Spectroscopy

• Observing spectral lines requires to make synthesis images at a large number ofclosely spaced frequencies

• This can be done by implementing a large number of multipliers to calculate thecorrelation function as a function of lag τ . Noting that each lag results in a phaseshift τν at frequency ν, V (u, v, τ ) is given by:

Vν(u, v, τ ) =

∫V (u, v, ν).ei2πτν.dν (26)

This Fourier transform can be inverted to retrieve the complex visibility V (u, v, ν)

• This is feasible because the complex visibility V (u, v, ν) of the source varies ontimescales (dominated by Earth rotation) which is significantly larger than the lagstep τ = 1/∆ν.

• This additional lag machinery allows coarse compensation of the geometric delay.Compensation of the fine delay ∆τ = mod(τg, 1/∆ν) can be done by a simple phaseslope as function of frequency.

• For n lags, the spectral channel width is ∆ν/n and the low loss delay trackingcondition δν∆τ 1 is satisfied if n 1.


In summary...

In a heterodyne interferometer

1. Sign of Phase depends on sideband conversion

2. Frequency is the time derivative of the Phase

3. Finite Bandwidth implies delay tracking

4. a Delay error produces a linear phase slope as function of frequency

5. Fringe Stopping makes life easier, but implies a Complex Correlator

6. High resolution spectroscopy is natural

7. Delays can be implemented digitally

8. a Complex Correlator directly measures the Visibility


Instrumental Effects

• Delay tracking and fringe stopping require good knowledge of the geometrical delay,i.e. of the baseline coordinates.

• Accurate phase lock systems are required to control φLO

• Antenna deformations (e.g. thermal expansion) may change the geometrical delay.Proper control of the antenna focus is required.

• The atmosphere may introduce a variable, random delay above each antenna. Notethat this is a delay, although it is often referred to as a phase.

• and of course, there is noise...


Calibration: Why ?

• φLO may vary with time −→ Phase Calibration

• The LO power may vary with time −→ Amplitude Calibration

• The atmospheric transparency may vary with time −→ Amplitude Calibration

• The atmosphere introduces a non-geometric delay, because of varying water vapor−→ Phase Calibration

• The geometric delay must be known −→ Baseline Calibration

• The instrumental delay must be known −→ Delay Calibration

• The amplitude and phase response as f (ν) must be known −→ Bandpass Calibration


Calibration: How ?

• Observe sources for which you know the answer (i.e. the visibility)

• Baseline: on an ensemble of point sources of known direction (since φ = b.s/c,you can determine b if you have an ensemble of s).

• Bandpass and delay: on a strong continuum point source

• Atmospheric transparency: from atmospheric models

• Amplitude: observe point source of known flux

• LO phase: observe point sources regularly

• Atmospheric phase: sorry, you cannot...


Atmospheric Phase

• Can not (usually) be calibrated because

– Time scale is short (typically 100 sec)– Antennas cannot re-point so quickly

• However, atmospheric phases are usually small enough, i.e. much less than 1 radian.

• More precisely

– the phase error is ∝ ν, because the atmosphere actually introduces a varyingpathlength δPath

– the pathlength fluctuations goes as D0.8−0.3

– up to an outer scale of D ≈ 1000− 2000 m– the timescale is simply D/vwind (frozen atmosphere model), typically 30 seconds

for 300 m.– δPath(300 m) / 200− 300µm, λ.– this result in an the atmospheric seeing of about 0.3 – 1′′, nearly frequency

independent.

• Could be predicted by Water Vapor Monitoring, because water is (by large) thedominant refractive component of the atmosphere.


The notion of Noise Temperature, Brightness and Flux• The output power of any electronic device is simply given by its noise

temperature. This is the Nyquist’s relation (W = k.T.∆ν).

• An antenna is an electronic device. Its impedance can be derived from its opticalcharacteristics. We define its equivalent noise temperature Tsys as the physicaltemperature of a resistor with same impedance delivering the same output power.

• The power received from a source of flux density Sν (Watt.m2.Hz−1) by an antennaof collecting area A (m2), aperture efficiency ηA over bandwidth ∆ν (Hz) is

Ps =1

2ηAA∆νSν = kTA∆ν (27)

• TA is the antenna temperature of the source

Sν =2k

ηAATA = J TA

• In an integration time t, there are about 2∆ν independent samples, giving a signal tonoise:

S/N = CTA

Tsys

1√∆νt

(28)

where C is a constant of order 1, depending on the details of receiving system.


• (Rayleigh-Jeans) brightness temperature Tb and Flux density Sν are related by thesimple integral

Sν = 2kTbΩb/λ2 (29)

Ωb being the solid angle and λ the wavelength (k the Boltzmann constant)

• and the antenna primary beam and aperture are related by the antenna equation

ΩPAeff = λ2 (30)

TA and Tb are similar quantities: TA = Tb for sources filling the antenna beam.

• Noise temperatures when referred to the aperture plane of the antenna aredirectly comparable to the (Rayleigh-Jeans) brightness temperatures of the sky.

• This allows a very simple and direct comparison between the sky emission and thetotal system noise. The noise equation is

∆Tb =Tsys√∆νt

(31)

where ∆ν is the bandwidth, t the integration time, Tsys the so-called systemtemperature and ∆Tb the resulting noise.


Sensitivity of an interferometer

• For a single baseline interferometer, the noise level is

σ =

√2kTsys

ηAηQA√

∆νt=

1√2

J Tsys

ηQ

√∆νt

(32)

where A is the collecting area of one antenna, and ηQ is the correlator efficiency(between 0.64 and 1, for (good) digital correlators). J is the Jy/K conversion factorfor one antenna.

• σ is√

2 worse than for an antenna of equal total collecting area, but no reference isrequired, so t can be 2× longer...

• This “loss” of sensitivity compared to a single-dish is due to the lack of analysis ofthe auto-correlation products.


• With n antennas, there are n(n− 1)/2 independent measurements, so

σ =2kTsys

ηAηQηφA√

n(n− 1)∆νt=

J Tsys

ηQηφ

√n(n− 1)∆νt

(33)

where the term ηφ is the decorrelation factor due to phase noise:

ηφ = e−(∆φ)2/2 (34)

with ∆φ the baseline-based phase noise.

• As Aeff = ηAA = λ2/ΩP and Sν = 2kTbΩS/λ2, Eq.33 simplifies too

∆Tb =Tsys

n√

∆νt

(θP

θS

)2

(35)

i.e. that of a fictitious antenna with the same total collecting area (nAeff , sameprimary beam θP, corrected for the beam dilution factor (θP/θS)

2, except forsmall losses ηQ and ηφ


Calibration of single-dish mm telescope

• So far, we had neglected atmospheric opacity...

• The noise power detected by the receiver is WA + Wrec, the sum of thepower received by the antenna and the noise generated by the receiver +transmission lines.

• The power received by the antenna is WA = ηWsky + (1− η)Wground, whereη is the forward efficiency (coupling to the sky)

• Using Nyquist’s relation (W = k.T.∆ν) we can convert that to temperatures

Tant = Trec + ηTsky + (1− η)Tground (36)

Tsky = e−τTbgg + (1− e−τ)Tatm + e−τT ∗A (37)

where τ is the atmospheric opacity.

• T ∗A is the antenna temperature of the astronomical source. It is equal to the

brightness temperature of the source if the source fills the primary beam.

• Knowledge of Trec and η and measurement of the output power allows determinationof the atmospheric opacity τ when a reasonable model for Tatm can be used.


Antenna and System Temperatures• The noise temperature Tant refers to the aperture plane of the antenna. If we instead

refer to the brightness temperature scale of the source, we can express the systemtemperature

Tsys =eτ

η(Trec + ηTsky + (1− η)Tground) (38)

• The System Temperature Tsys is the equivalent noise temperature of all theinstrumental system including the atmosphere emission. It is directly comparable tothe source brightness temperature.

• Best values of Tsys for mm radio telescopes (e.g IRAM 30-m or PdBI) are:

– Tsys(3mm or 90 GHz) ' 60− 100K

– Tsys(1.3mm or 230 GHz) ' 250− 350K

• Typical values of T ∗A observed with single-dish 15-m antennas (PdBI) are:

– T ∗A (CO J=1-0 in dark cloud) ' 5− 10 K

– T ∗A (CO J=2-1 in Keplerian disk) ' 0.5 K

– T ∗A (CO in high-z galaxy) ' 5 mK

• Ratio: T ∗A/Tsys ∼ 10−2 − 10−6


Atmospheric transparency at mm and submm wavelengths.


Sensitivity of current mm arrays

• The antenna Jy/K gain is J = 2k/(Aeff) ' 35 Jy/K, at 1.3mm for the ALMAantennas, 30 for the IRAM antennas

∆Sν =J · Tsys

ηQ

√2Nbase∆νtint

(39)

– Tsys = 300 K, Nbase = n(n− 1)/2 = 10, ηQ = 0.6 (correlator loss & decorrelationfactor)

• In continuum, ∆ν = 500 MHz

– 1 hour integration time −→ 2.8 mJy/beam– Best: 24 hour integration time −→ 0.5 mJy/beam

• For spectral lines of line-width ∆ν = 100 kHz (or ∆V = 0.1 km/s)

– 10 hours integration time −→ 60 mJy/beam– therefore 1.4 K at 1′′ of angular resolution at 230 GHz (140 K at 0.1′′...)

• ALMA will be 30 to 100 times more sensitive (2016 baselines instead of 10, betterTsys...)


Orders of magnitude

∆ν = 500 MHz

t = 6 h

(∆νt)−12 = 3 10−7

A = 1000 m2

ηA = 0.5

Sν = 1 mJy

Tsys = 200 K

leads to a 3σ detection, and a total energy received from the source of 5 10−14 J.(the received energy in 1 Million year would be about the kinetic energy of a fallingsnowflake...)


• Noise properties for 1 baseline vary with Signal-to-Noise ratio

• On the amplitude & flux density

S << σ

σA ' σ√

2− π2

(1 +

(S2σ

)2)

<S> ' σ√

π2

(1 +

(S2σ

)2) (40)

S >> σ

σA ' σ

<S> ' S(41)

• On the phase

S << σ

σφ ' π√3

(1−

√9

2π3Sσ

)(42)

S >> σσφ ' σ

S(43)

• Source detection is much easier on the phase than on the amplitude, since forS/N ∼ 1, σφ = 1radian = 60.


Conclusions

• mm interferometry is not so difficult to understand

• even if you don’t, the noise equation is all you need

• the noise equation

∆Tb =Tsys

ηn√

∆νt

(θP

θS

)2

(44)

allows you to check quickly if a source of given brightness Tb can be imaged at agiven angular resolution θS and spectral resolution ∆ν (n is the number of antennas,θP their primary beam width, and η an efficiency factor of order 0.5)

• Tsys is easy to guess: the simplistic value of 1 K per GHz of observing frequency is agood enough approximation in most cases.

• and you know Tb because you know the physics of your source!

• that is (almost) all you need to decide on the feasibility of an observation...


Documents

S.Guilloteau (IRAM/ESO) ALMA European Project Scientist · S.Guilloteau (IRAM/ESO) ALMA European Project Scientist Bibliography: • “Synthesis Imaging”. ... Basic principle •